PANEL Procedure

Linear Hypothesis Testing

Consider a linear hypothesis of the form bold upper R bold-italic beta equals bold q, where bold upper R is upper J times upper K and bold q is upper J times 1. The Wald test statistic is

chi Subscript normal upper W Superscript 2 Baseline equals left-parenthesis bold upper R ModifyingAbove bold-italic beta With caret minus bold q right-parenthesis Superscript prime Baseline left-parenthesis bold upper R ModifyingAbove bold upper V With caret bold upper R prime right-parenthesis Superscript negative 1 Baseline left-parenthesis bold upper R ModifyingAbove bold-italic beta With caret minus bold q right-parenthesis

where ModifyingAbove bold upper V With caret is the estimated variance of ModifyingAbove bold-italic beta With caret.

In simple linear models, the Wald test statistic is equal to the F test statistic

upper F equals StartFraction left-parenthesis normal upper S normal upper S normal upper E Subscript r Baseline minus normal upper S normal upper S normal upper E Subscript u Baseline right-parenthesis slash upper J Over normal upper S normal upper S normal upper E Subscript u Baseline slash d f Subscript e Baseline EndFraction

where normal upper S normal upper S normal upper E Subscript r is the restricted error sum of squares, normal upper S normal upper S normal upper E Subscript u is the unrestricted error sum of squares, and d f Subscript e is the unrestricted error degrees of freedom.

The F statistic represents a more direct comparison of the restricted model to the unrestricted model. Comparing error sums of squares is appealing in complex models for which restrictions are applied not only during the final regression but also during intermediate calculations.

The likelihood ratio (LR) test and the Lagrange multiplier (LM) test are derived from the F statistic. The LR test statistic is

chi Subscript normal upper L normal upper R Superscript 2 Baseline equals upper M ln left-bracket 1 plus StartFraction upper J upper F Over upper M minus upper K EndFraction right-bracket

The LM test statistic is

chi Subscript normal upper L normal upper M Superscript 2 Baseline equals upper M left-bracket StartFraction upper J upper F Over upper M minus upper K italic plus upper J upper F EndFraction right-bracket

The distribution of these test statistics is chi squared with J degrees of freedom. The three tests are asymptotically equivalent, but they possess different small-sample properties. For more information, see Greene (2000, p. 392) and Davidson and MacKinnon (1993, pp. 456–458).

Only the Wald is changed when a heteroscedasticity-corrected covariance matrix estimator (HCCME) is selected. The LR and LM tests are unchanged.

Last updated: June 19, 2025